Parent to vertex unwanted offset
Short answer as already answered here  Wrapping geometry object animation 

If you start with a face setting like this, you can rotate 72° to obtain the other faces:



Now if you set the snapping like this:



And use an empty as active element, you can place the face next to the others as you need:



Finally you obtain this:



From that you'll need (I describe this quickly as this is not the direct subject):


to join the bones, 
join the faces (and remove double)
parent them with empty groups and assign the needed vertices to the corresponding groups manually


Now, as the bones are all along their local axis (along Y), you can rotate them all around X using 'individual origins':



So you can rotate them all around their local X axis, using RXX (twice X for local axis):



Last step: the dodecahedron as an angle of arcTan(2) = 63.434949°. So we need 
to set a keyframe for this rotation value.